= Connection-independent curvature class of a line bundle
{title2=$[F_A]\in H^2_{\mathrm{dR}}(M;k)$}
For a real or complex <line bundle>, its <endomorphism bundle> is canonically trivial by scalar multiplication. The <Bianchi identity> therefore reduces to $dF_A=0$. The difference of two <connections on a vector bundle> is a global endomorphism-valued one-form, which here is an ordinary scalar one-form $a$. Since scalar one-forms have vanishing wedge square, the <curvature difference formula> reduces to $F_{A+a}-F_A=da$. Thus the <de Rham cohomology> class is independent of the <connection on a vector bundle>.
Back to article page
